The problem asks to find $f\left(\frac{2b}{3}\right)$ given that $f(x) = x^2 - 3$.

Step-by-step solution:

1. Substitute $\frac{2b}{3}$ for $x$ in the given function $f(x) = x^2 - 3$:

   $f\left( \frac{2b}{3} \right) = \left( \frac{2b}{3} \right)^2 - 3$

2. Square $\frac{2b}{3}$:

   $\left( \frac{2b}{3} \right)^2 = \frac{(2b)^2}{3^2} = \frac{4b^2}{9}$

3. Substitute this into the function:

   $f\left( \frac{2b}{3} \right) = \frac{4b^2}{9} - 3$

4. Simplify the expression by rewriting 3 as a fraction with denominator 9:

   $f\left( \frac{2b}{3} \right) = \frac{4b^2}{9} - \frac{27}{9}$

5. Combine the terms:

   $f\left( \frac{2b}{3} \right) = \frac{4b^2 - 27}{9}$

Thus, the correct answer is:

$\frac{4b^2 - 27}{9}$

The answer matches the last option in the image.

Would you like more details on this, or do you have any further questions?

Here are 5 related questions to practice:

1. If $f(x) = x^2 - 5$, what is $f\left(\frac{3b}{4}\right)$?
2. Find $f(2b)$ for $f(x) = x^2 - 3$.
3. How do you evaluate $f(x) = 4x^2 - 1$ at $x = 2b + 1$?
4. What is the result of $f\left(\frac{b}{2}\right)$ for $f(x) = x^2 - 6$?
5. Simplify $\left( \frac{3b}{5} \right)^2 - 2$.

Tip: When substituting variables into functions, remember to square or cube them as required, then simplify carefully.